1 Introduction↩︎

The evaluation of multi-loop Feynman integrals is one of the most important and challenging tasks in quantum field theory. In its modern workflow, the integration-by-parts (IBP) reduction [1], [2] is often a bottleneck step in many frontier problems. In recent decades, various methods and software have been developed to perform IBP reduction [3]–[62]. Still, the reduction is difficult for integrals with large number of loops, external legs, or mass parameters.

In the effort of finding better algorithms for IBP reduction, the generalized unitarity cuts (GUC) [23], [24], [63]–[72] is proved to be useful. By setting some of the propagators on-shell, the integral families are usually simplified since plenty of its sectors become zero sectors. Based on the conjuncture that GUC keeps the IBP relation unchanged, since it does not affect the integrand and the boundary, the so-called spanning cuts method [73], [74] was introduced¹ to simplify the IBP reduction. With a set of cuts that covers all the non-zero sectors of the family, a huge reduction problem can be decomposed to plenty of much simpler ones. As reported in [77], for example, the spanning cuts method is vital for breaking through the computational bottlenecks of nowadays’s frontier problems. This method was also implemented in many public software [51], [54], [57], [61], [78].

Despite its importance, the spanning cuts method itself suffers from the so-called consistency problem: As what was believed, the IBP reduction coefficients of an integral should agree with each other under different cuts, inferred by the conjecture that IBP relations are unchanged under GUC. This is a requirement called the consistency condition. However, many studies [51], [57], [78] have reported violations of this condition.

Since its discovery, people are researching the cause of this problem, as well as finding solutions to it. In Ref. [51], [78], the consistency problem was reported to be having correlation to the so-called magic relations [19], [32], [79], which is a kind of IBP relation induced from a super sector but contains integrals only in its sub sectors. Ref. [51] has also designed a variation of the spanning cuts method to avoid the influence of magic relations.

Even with these studies, the cause of this problem is still unrevealed. We still need to know what exactly happened in the cases that the consistency condition breaks. It is still mysterious why we see some magic relations breaks under the cuts, knowing that they can be expressed as linear combination of ordinary IBP relations, which we believe to be kept under the cuts.

In this paper, we will provide our answer to the question. We claim that the conjecture that IBP relations is unchanged under the cut is not one-hundred-percent correct if we drop out the hidden terms in the IBP relations. The hidden terms are terms proportional to the vanishing Feynman prescriptions. We are used to erase the hidden terms in the IBP relations since we believe that they vanish. However, in this paper, we will show that, under certain choice of cuts, they do not vanish. In these cases, the vanishing prescription parameters are canceled by pinch singularities introduced by the cuts, making the hidden terms in fact finite. If we blindly drop them out, the IBP relations will appear to be violated. We will show with an example that, if we keep those hidden terms, the IBP relation still holds under these cuts.

This paper is organized as follows. In Section 2, we review some concepts. In Section 3, we provide a simple but typical example, the one-loop massive bubble, to show how hidden terms spoils the IBP relations under the cut. We also raise a concept named hidden relations to help us detect the possible pinch singularities under the cut. In Section 4, we provide an algorithm and its implementation to find linear hidden relations. In Section 5, we provide a more complicated two-loop example without linear hidden relation, where inconsistencies still occur. In Section 6, we summarize this paper.

2 Review of concepts and methods↩︎

2.1 Feynman integrals and sectors↩︎

A generic Feynman integral with \(L\) loops is defined as, \[I_{\alpha_1,\cdots,\alpha_n}=\int \prod_{j=1}^L \frac{{\rm d}^D l_j}{i \pi^{D/2}} \, \frac{1}{\prod_{i=1}^n P_i^{\alpha_i}},\] where \(\alpha_i \in \mathbb{Z}\) are integer indices, and the propagators \(P_i\) are linear or quadratic polynomials in the loop momenta \(l_j\) and external momenta \(p_i\), given by \[\begin{gather} P_i = \sum_{j ,k=1}^L A_i^{jk} \, l_j \cdot l_k + \sum_{j=1}^L B_i^j \cdot l_j + E_i. \end{gather}\] Here, the coefficients \(A_i^{jk}\) and \(B_i^j\) depend on the external kinematics, and \(E_i\) is a scalar function of external momenta. One should keep in mind that there are Feynman prescription parameters in the denominators, which we did not explicitly write down here.

The sector of an integral is defined to label which of the propagators appears in the denominator of the integrand. Exlicitly,

\[{\text{Sec}}(I_{\alpha_1,\cdots,\alpha_n})=\{i|\alpha_i>0\}\]

2.2 IBPs and spanning cuts↩︎

The integration-by-parts (IBP) identities [1], [2] are \[\begin{gather} 0 = \int \prod_{j=1}^L \frac{{\rm d}^D l_j}{i \pi^{D/2}} \, \frac{\partial}{\partial l_k^\mu} \left( \frac{v_k^\mu}{\prod_{i=1}^n P_i^{\alpha_i}} \right), \label{IBP-relation} \end{gather}\tag{1}\] where \(v_k^\mu\) is a vector linear in loop or external momenta.

The Feynman integrals under a certain cut is defined as

\[\begin{gather} I_{\alpha_1,\cdots,\alpha_n}|_{\{{\mathcal{C}}\}\text{-cut}} = \int \prod_{j=1}^L \frac{{\rm d}^D l_j}{i \pi^{D/2}} \, \frac{1}{\prod_{i \notin \mathcal{C}} P_i^{\alpha_i}} \, \prod_{k \in \mathcal{C}} \delta(P_k). \end{gather}\]

if \(\alpha_k=1\) for \(k\in{\mathcal{C}}\). Where \({\mathcal{C}}\) is a sub set of \(\{1,2,\cdots,n\}\). If \(\alpha_k\neq1\), it is clearer to define GUC in the Baikov representation [80]–[82], where an integral is expressed as \[\label{eq:Baikov32rep} I_{\alpha_1,\cdots,\alpha_n}=J\int_\Omega{\rm d}z_1 \cdots {\rm d}z_n P^\gamma \frac{1}{z_1^{\alpha_1} \cdots z_n^{\alpha_n}},\tag{2}\] where \(J\) and \(\gamma\) are constants. \(P\) is a polynomial of \(z_i\)’s, and \(\Omega\) is a boundary where \(P=0\). The integrals under cut are \[\label{eq:32C-cut} I_{\alpha_1,\cdots,\alpha_n}|_{{\mathcal{C}}\text{-cut}}:=J\int_{\Omega_{\mathcal{C}}}\prod_{i\notin{\mathcal{C}}} {\rm d}z_i\prod_{i\in{\mathcal{C}}}\oint_0 {\rm d}z_i\frac{P^\gamma}{{z_1}^{\alpha_1}\cdots {z_n}^{\alpha_n}},\tag{3}\]

With this definition, we see that \[{\mathcal{C}}\not{\subseteq}{\text{Sec}}({I_{\alpha_1,\cdots,\alpha_n}})\Rightarrow {I_{\alpha_1,\cdots,\alpha_n}}|_{\{{\mathcal{C}}\}\text{-cut}}=0\]

Thus, we can produce plenty of zero-sectors using GUC, such that all the integrals belonging to the sector vanish. For some reduction problems, this significantly reduces the cost of the computation.

Bases on this, the spanning cuts idea is as follows. Firstly, we find out a set of cuts \[\{{\mathcal{C}}_i\},\] such that for any non-zero sector \(S\) in a family, \[\{{\mathcal{C}}_i\}\cap {\text{Subsets}}(S)\neq\{\}.\] Then, we perform IBP reductions for a certain integral \(I\) on the spanning cuts, which should be faster, to get \[I|_{\{{\mathcal{C}}_i\}\text{-cut}}=\sum_j c_j^i I_j|_{\{{\mathcal{C}}_i\}\text{-cut}},\] where the R.H.S is the linear combination of master integrals under the cuts.

We then use the above results to reconstruct the IBP reduction result of the original problem, without cuts. We label it as \[I=\sum_j c_j I_j.\] If we admit the conjecture that IBP relations are unchanged under cuts, we have \[\label{eq:spc32merge} c_j^i=\begin{cases} c_j,\, &\text{if } {\mathcal{C}}_i\subseteq{\text{Sec}}(I_j)\\ 0,\, &\text{if } {\mathcal{C}}_i\not\subseteq{\text{Sec}}(I_j) \end{cases}.\tag{4}\] We can use the above to reconstruct \(c_j\). We pick up one \({\mathcal{C}}_i\) such that \({\mathcal{C}}_i\subseteq{\text{Sec}}(I_j)\), then we have \[c_j=c_j^i.\] When there are multiple choices, we require that all the corresponding \(c_j^i\) of these choices should be the same. Otherwise, 4 does not hold. This is called the consistency condition of the spanning cuts method. However, as stated, this condition is violated in many observations. This leads to the consistency problem studied in this paper.

3 A simple example with inconsistency↩︎

In this section, we provide one simple but typical example where the inconsistency occurs.

3.1 Example description↩︎

We consider the one-loop bubble with massive propagators and massless external legs shown in Figure 1. The propagators are defined as \[P_1=l^2-m^2,\, P_2=(l+p)^2-m^2.\] The external kinematic condition is \[p^2=0.\]

The integrals in this family is then \[I_{\alpha_1,\alpha_2}=\int \frac{{\rm d}^{D}l}{i \pi^{\frac{D}{2}}}\frac{1}{P_1^{\alpha_1}P_2^{\alpha_2}}.\]

We pick up one of the IBP relations as \[\label{eq:32IBP32example32im32bubble} 0=\int \frac{{\rm d}^{D}l}{i \pi^{\frac{D}{2}}}\frac{\partial}{\partial l^{\mu}}\frac{p^\mu}{P_1P_2} =(I_{0,2}-I_{2,0})\tag{5}\] If we apply \(\{1\}\text{-cut}\) on the equations, we have \[I_{0,2}|_{\{1\}\text{-cut}}=0\] If we admit the conjecture that GUC keeps the IBP relation 5 unchanged, we will have \[I_{2,0}|_{\{1\}\text{-cut}}=0\] However, this is not true. One can directly compute it as

\[\label{eq:1-cut32spec32calculation} \begin{align} I_{2,0}|_{\{1\}\text{-cut}}&=-2\pi i\int \frac{{\rm d}^{D}l}{i \pi^{\frac{D}{2}}}\delta^\prime((l^0)^2-|\vec{l}|^2-m^2)=2\pi i\int \frac{{\rm d}^{D-1}\vec{l}}{i \pi^{\frac{D}{2}}}\int{\rm d}\big( \frac{{\rm d}l^0}{{\rm d}P_1}\big)\delta((l^0)^2-|\vec{l}|^2-m^2)\\ &=\int \frac{{\rm d}^{D-1}\vec{l}}{i \pi^{\frac{D}{2}}}\int\frac{-2\pi i}{2 (l^0)^2}\delta((l^0)^2-|\vec{l}|^2-m^2){\rm d}l^0=\int \frac{{\rm d}^{D-1}\vec{l}}{i \pi^{\frac{D}{2}}}\frac{-2\pi i}{2(|\vec{l}|^2+m^2)^{\frac{3}{2}}}\\ &=-2(m^2)^{\frac{D-4}{2}}\Gamma(2-\frac{D}{2})\neq0 \end{align}\tag{6}\]

The contradiction indicates that the conjecture is not true in this case. We will find the “bug” for this example in the next sections.

3.2 The cuts with Feynman prescription parameters kept↩︎

The contradiction in Section 3 can be addressed more clearly if we keep the Feynman prescription parameters. Specifically, for an integral family with \(n\) propagators \(P_1,\cdots,P_n\), we define \[D_k= P_k - i \eta_k, \quad \Bar{D}_k= P_k +i \eta_k\] where \(k=1,2,\cdots , n\) and \(\eta_k\) will be taken as \(0^+\) after all computation finished. We rewrite the Feynman integrals \(I_{\alpha_1,\cdots,\alpha_n}\) in a finer way as \[I_{\alpha_1,\cdots\alpha_n}=\lim_{\text{all }\eta_k\to0^+}\mathcal{I}_{\alpha_1,\cdots\alpha_n}\] where \[\mathcal{I}_{\alpha_1,\cdots\alpha_n}=\int \frac{{\rm d}^{D}l_1}{i \pi^{\frac{D}{2}}}\cdots \frac{{\rm d}^{D}l_L}{i \pi^{\frac{D}{2}}}\frac{1}{D_1^{\alpha_1}\cdots D_n^{\alpha_n}}\]

We can define GUC through the “mirroring” the poles of the corresponding propagators. To do this, we define an \(s\)-mirrored integral, where \(s\) is a set of indices, as

\[I_{\alpha_1,\cdots\alpha_n}|_{s\text{-Mir}}:=\lim_{\text{all }\eta_j\to0^+}{\mathcal{I}}_{\alpha_1,\cdots\alpha_n}|_{s\text{-mir}},\] where \[f(\eta_1,\cdots,\eta_n)|_{s\text{-mir}}:=f(\eta_1,\cdots,\eta_n)|_{\eta_k \to -\eta_k,\, \text{for all } k\in s}\] for an expression \(f(\eta_1,\cdots,\eta_n)\) containing the \(\eta\)’s. Notice that the mirror label with capital and lowercase of “M” have different definitions.

With the above, the GUC can be defined as

\[I_{\alpha_1,\cdots\alpha_n}|_{\mathcal{C}\text{-cut}}=\lim_{\text{all }\eta_j\to0^+}\sum_{s \in \text{subsets of }\mathcal{C}}(-1)^{|s|}{\mathcal{I}}_{\alpha_1,\cdots\alpha_n}|_{s\text{-mir}},\] where \(|s|\) is the number of elements in \(s\). For example, if we cut one propagator, i.e. \(s=\{k\}\), the above simplifies to \[I_{\alpha_1,\cdots\alpha_n}|_{\{k\}\text{-cut}}=\lim_{\text{all }\eta_j\to0^+}\int \frac{{\rm d}^{D}l_1}{i \pi^{\frac{D}{2}}}\cdots \frac{{\rm d}^{D}l_L}{i \pi^{\frac{D}{2}}}\Big(\frac{1}{D_1^{\alpha_1}\cdots D_n^{\alpha_n}}-\frac{1}{D_1^{\alpha_1}\cdots D_n^{\alpha_n}}\Big|_{\eta_k \to -\eta_k}\Big).\] This is consistent with what we are familiar with, if we admit the following relation² \[\lim_{\eta_k\to 0^+}(\frac{1}{D_k}-\frac{1}{\Bar{D}_k})=2 \pi i \delta(P_k)\]

3.3 The cause for the inconsistency↩︎

With the Feynman prescription kept, the IBP relation in 5 is \[\label{eq:full32IBP32of32imbubble32example} 0=\int \frac{{\rm d}^{D}l}{i \pi^{\frac{D}{2}}}\frac{\partial}{\partial l^{\mu}}\frac{p^\mu}{D_1D_2} =({\mathcal{I}}_{0,2}-{\mathcal{I}}_{2,0}-i(\eta_2-\eta_1)({\mathcal{I}}_{1,2}+{\mathcal{I}}_{2,1}))\tag{7}\] Applying \(\{1\}\text{-mir}\) to 7 , we have \[\label{eq:full32IBP32with321-mir32of32imbubble32example} 0=\int \frac{{\rm d}^{D}l}{i \pi^{\frac{D}{2}}}\frac{\partial}{\partial l^{\mu}}\frac{p^\mu}{\bar{D}_1D_2} =({\mathcal{I}}_{0,2}|_{\{1\}\text{-mir}}-{\mathcal{I}}_{2,0}|_{\{1\}\text{-mir}}-i(\eta_2+\eta_1)({\mathcal{I}}_{1,2}|_{\{1\}\text{-mir}}+{\mathcal{I}}_{2,1}|_{\{1\}\text{-mir}}))\tag{8}\]

In the above IBP relations, there are some terms that are proportional to Feynman prescription parameters, \(\eta_k\)’s, as \[-i(\eta_2+\eta_1)({\mathcal{I}}_{1,2}|_{\{1\}\text{-mir}}+{\mathcal{I}}_{2,1}|_{\{1\}\text{-mir}}))\] These terms are the so-called hidden terms in this paper. We will show that they are very closely related to the inconsistency problem. In the past, when we derive IBP relations, we always ignored hidden terms, since they are multiplied by \(\eta_k\)’s that will be taken to \({0^+}\). However, this is based on the assumption that the integrals, which multiplies \(\eta_k\)’s, are finite. We will show that this assumption is not always true after we apply “mirror” operations. For this example, we will show that \[{\mathcal{I}}_{1,2}|_{\{1\}\text{-mir}}+{\mathcal{I}}_{2,1}|_{\{1\}\text{-mir}}\sim \frac{1}{\eta_1+\eta_2}\] so that it cancels the “vanishing” coefficient \(\eta_1+\eta_2\) in its front, making the whole term non-vanishing.

3.3.1 Hidden terms without mirroring \(\eta_1\)↩︎

We can first look at the hidden terms in 7 . Without losing generality, we define \[\label{eq:l32specify} l=(l^0,l^1,\vec{z}^{\,(D-2)})\tag{9}\] \[\label{eq:p32specify} p=(|\vec{p}\,|,|\vec{p}\,|,\vec{0}^{(D-2)})\tag{10}\] Integrating over \(l^0\), we have \[{\mathcal{I}}_{1,2}+{\mathcal{I}}_{2,1}=\int\frac{{\rm d}^{D-1}{\vec{l}}}{i \pi^{\frac{D}{2}}}{\mathcal{B}}_{12},\] where \[\label{eq:B12} {\mathcal{B}}_{12}= \frac{i \pi r_{12}}{2 r_1^2 r_2^2 (|{\vec{p}}\,|^2-r_{12}^2)} \Big(1+\frac{2 |{\vec{p}}\,|^2}{|{\vec{p}}\,|^2-r_{12}^2}-\frac{r_{12}^2}{r_1 r_2}\Big)\tag{11}\] and \[r_1=\sqrt{|{\vec{l}}\,|^2+m^2+i \eta_1},\quad r_2=\sqrt{|{\vec{l}}+{\vec{p}}\,|^2+m^2+i \eta_2},\quad r_{12}=r_1+r_2.\] We expect that the integration of \(B_{12}\) over \(l^1\) is finite when \(\eta_1\) and \(\eta_2\) vanishes, because we have \[\label{eq:B1232limit} \lim_{l^1\to \pm\infty}{\mathcal{B}}_{12}|_{\eta_1=0,\eta_2=0} \sim \pm\frac{3i\pi }{4(l^1)^5}\to 0,\tag{12}\] and \[r_{12}>|{\vec{p}}\,|, \quad \text{when }\eta_1,\eta_2=0,\quad\forall l^1\in(-\infty,+\infty).\]

We can directly compute the integration over \(l^1\) as³ \[\label{eq:IntB1232definite} {\mathcal{I}}_{1,2}+{\mathcal{I}}_{2,1}=\int\frac{{\rm d}^{D-2}\vec{z}}{i \pi^{\frac{D}{2}}}\frac{i \pi }{ ( |\vec{z}\,|^2+m^2+i\eta _1)( |\vec{z}\,|^2+m^2+i\eta _2)}\tag{13}\] This is indeed finite. Thus, the term \((\eta_2-\eta_1)({\mathcal{I}}_{1,2}+{\mathcal{I}}_{2,1})\) vanishes after taking \(\eta_1,\eta_2=0^+\).

3.3.2 Hidden terms with mirroring \(\eta_1\)↩︎

For the hidden terms in 8 , the story is very different compared to the previous case. Again, using the gauge in 9 and 10 , and integrating over \(l^0\) we have

\[{\mathcal{I}}_{1,2}|_{\{1\}\text{-mir}}+{\mathcal{I}}_{2,1}|_{\{1\}\text{-mir}} = \int\frac{{\rm d}^{D-1}{\vec{l}}}{i \pi^{\frac{D}{2}}}{\mathcal{B}}_{{\underline{1}}2}\] where \[\label{eq:Bu12} {\mathcal{B}}_{{\underline{1}}2}= \frac{i \pi r_{{\underline{1}}2}}{2 r_{\underline{1}}^2 r_2^2 (|{\vec{p}}\,|^2-r_{{\underline{1}}2}^2)} \Big(1+\frac{2 |{\vec{p}}\,|^2}{|{\vec{p}}\,|^2-r_{{\underline{1}}2}^2}-\frac{r_{{\underline{1}}2}^2}{r_{\underline{1}}r_2}\Big)\tag{14}\] and \[r_{\underline{1}}=-\sqrt{|{\vec{l}}\,|^2+m^2-i \eta_1},\quad r_2=\sqrt{|{\vec{l}}+{\vec{p}}\,|^2+m^2+i \eta_2},\quad r_{{\underline{1}}2}=r_{\underline{1}}+r_2\] This time, the integration of \(B_{{\underline{1}}2}\) over \(l^1\) is no longer convergent when \(\eta_1,\eta_2\to {0^+}\), because \[\label{eq:32Bu1232limit} \lim_{l^1\to \pm\infty}{\mathcal{B}}_{{\underline{1}}2}|_{\eta_1=0,\eta_2=0} = \pm\frac{i\pi }{|{\vec{p}}\,|(|{\vec{z}}\,|^2+m^2)^2}\neq0.\tag{15}\] Consequently, we should keep \(\eta_1\) and \(\eta_2\) as regulators of the divergences while computing the integral. Otherwise, the expression becomes ill.

We now compute the integration of \({\mathcal{B}}_{\underline{1}2}\) over \(l^1\) with \(\eta\)’s kept. The result is⁴

\[\label{eq:IntBu1232definite} {\mathcal{I}}_{1,2}|_{\{1\}\text{-mir}}+{\mathcal{I}}_{2,1}|_{\{1\}\text{-mir}} = \int\frac{{\rm d}^{D-2}\vec{z}}{i \pi^{\frac{D}{2}}}\frac{- \pi (2 (| \vec{z}\, | ^2+m^2)-i\eta _1+i\eta _2)}{(\eta _1+\eta _2) (| \vec{z}\, | ^2+m^2-i \eta _1) (| \vec{z}\, | ^2+m^2+i \eta _2)}\tag{16}\]

The above results is indeed divergent with a pole proportional to \(\frac{1}{\eta_1+\eta_2}\). This exactly cancels the “vanishing” coefficient \(\eta_1+\eta_2\) in its front. Thus, the term is finite: \[\label{eq:conpemsate32term} \lim_{\eta_1,\eta_2\to0^+}i({\eta_1+\eta_2})({\mathcal{I}}_{1,2}|_{\{1\}\text{-mir}}+{\mathcal{I}}_{2,1}|_{\{1\}\text{-mir}}) = \int\frac{{\rm d}^{D-2}\vec{z}}{i \pi^{\frac{D}{2}}}\frac{-2\pi i }{ (| \vec{z}\, | ^2+m^2)}=-2(m^2)^{\frac{D-4}{2}}\Gamma(2-\frac{D}{2})\tag{17}\] In the above, we took \(\eta_1\) and \(\eta_2\) as \(0^+\) before the integration over \({\vec{z}}\), since the integral is convergent after the pole \(\frac{1}{\eta_1+\eta_2}\) is canceled. With the result in 17 , we can subtract 7 and 8 while taking \(\eta_1, \eta_2=0^+\), to obtain \[\label{eq:imbubble32example32final32correct32IBP321-cut} 0=I_{0,2}|_{\{1\}\text{-cut}}-I_{2,0}|_{\{1\}\text{-cut}}-2(m^2)^{\frac{D-4}{2}}\Gamma(2-\frac{D}{2})\tag{18}\] Since \(I_{0,2}|_{\{1\}\text{-cut}}=0\), 18 agrees very well with 6 . Thus, we can conclude that the hidden terms that we have previously ignored is the cause of the “inconsistency” of IBP relations, at least in this example.

3.4 Comments↩︎

The inconsistency in this example comes from the fact that the two propagators \(P_1\) and \(P_2\) are identical in the sense of integration over loop momenta, although they are different apparently. This kind of identical relation is not the symmetry relation such that \(I_{\alpha_1,\alpha_2}=I_{\alpha_2,\alpha_1}\). It instead leads to an even stronger relation such that \(I_{\alpha_1,\alpha_2}=I_{\alpha_1+\alpha_2,0}\). This can be interpreted by the fact that the integral family depends only on \(m^2\), setting \(p\mapsto \lambda p\) does not change the integral. When we set \(\lambda=0\), \(P_2\) comes to \(P_1\). This leads to divergence while taking \(\{1\}\text{-cut}\) since, for example \[\lim_{\eta_1\to 0^+}({\mathcal{I}}_{\alpha_1,\alpha_2}|_{\{1\}\text{-cut}})\propto\int {\rm d}^{D}l \frac{\delta^{(\alpha_1-1)}(P_1)}{(P_1-i \eta_2)^{\alpha_2}} = \int {\rm d}^{D}l \frac{\delta^{(\alpha_1-1)}(P_1)}{(P_1-i \eta_2)^{\alpha_2}}\propto\frac{1}{\eta_2^{\alpha_1+\alpha_2-1}}\]

This is consistent with what we observed in 16 after taking \(\eta_1\to0^+\). Notice that, the singularity comes from the contribution where the poles \(-i \eta_1\) and \(i \eta_2\) located on the different side of the real axis, introducing a pinch singularity taking the limit \(\eta\to0\).

In general, for a sector \(S\), if the integral \(I_{\alpha_1,\cdots,\alpha_n}\) on this sector is unchanged after we do the following replacement \[\label{eq:32hidden32relation} P_i= f(P_1,\cdots,P_{i-1},P_{i+1},\cdots,P_n)\tag{19}\] in the integrand, where \(f\) satisfies \(f(0,0,\cdots,0)=0\), cutting the propagators other than \(P_i\) will introduce a divergence related to the Feynman prescriptions. These divergence could cancel with the coefficients in their front which vanishes when the Feynman prescriptions are set to \({0^+}\). In this paper, we call the relations like 19 as hidden relations.

4 Searching for linear hidden relations↩︎

As commented in Section 3.4, the existence of hidden relations may lead to hidden terms in IBP relations, making the IBP breaks (if the hidden terms are erased) under the cuts. Thus, finding these relations is helpful to detect possible inconsistencies when using the spanning cuts method. In this section, we demonstrate an algorithm to find linear hidden relations using Feynman parameterization.

To derive Feynman parameterization, one will introduce a linear combination of Feynman parameters and propagators as \[\label{eq:32FP32denomenator} \sum_{i=1}^n x_i P_i.\tag{20}\] If a linear hidden relation exists as \[P_i= \sum_{j=1,j\neq i}^n c_j P_j,\] 20 becomes \[\sum_{j=1,j\neq i}^n (x_j+c_j x_i) P_j.\] Consequently, the Symanzik polynomials (or \(G=U+F\), the Lee-Pomeransky polynomial [18], [20], [83]) should rely on \(n-1\) variables, i.e. \(x_j+c_j x_i\) for \(j\neq i\). Then, we will have \[\label{eq:32LP32pol32tangent32relation} \Big(-\frac{\partial}{\partial x_i}+\sum_{j=1,j\neq i}^n c_j \frac{\partial}{\partial x_j}\Big)G=0\tag{21}\] Equivalently, we can find the linear hidden relations by search for annihilating differential vectors shown in 21 . The algorithm is shown as Algorithm 2.

We provide an implementation of this algorithm in HiddenRelations.wl in the following GitHub repo.

As an example, we applied Algorithm 2 to the following integral family. Its propagators are \[\begin{align} &P_1=l_1^2-m^2,\quad P_2=(l_1+p_1)^2-m^2,\quad P_3=(l_1+p_1+p_2)^2-m^2,\\ &P_4=(l_2-p_1-p_2)^2-m^2,\quad P_5=(l_2+p_4)^2-m^2,\quad P_6=(l_2)^2-m^2,\\ &P_7=(l_1+l_2)^2-m^2,\quad P_8=(l_1+p_4)^2,\quad P_9=(l_2+p_1)^2. \end{align}\] The kinematic conditions are \[\begin{align} &p_1^2=p_2^2=p_4^2=0,\quad p_1\cdot p_2=\frac{s}{2},\quad p_1\cdot p_4=\frac{t}{2},\quad p_2\cdot p_4=\frac{-s-t}{2}. \end{align}\]

After searching all non-zero sectors, we found in total 65 sectors with annihilating differential vectors shown in 21 . We demonstrate some representative ones among them in Table 1. The corresponding diagrams are shown in Figure 3.

The cause of the inconsistencies is understandable for the sub diagram in Figure [subfig:imdb124] because it is factorized with a factor of massive bubble shown in Figure 1. The sub diagram in Figure [subfig:imdb2345] is similar. It has two factorized massive bubbles. So, it has two hidden relations. The sub diagram shown in Figure [subfig:imdb1267] does not have such massive bubble factor, but one can also set \(p_1\to 0\) since the diagram has only one massless external line. After doing so, \(P_1\) and \(P_2\) become identical, introducing divergence when cutting one of them. The sub diagram shown in Figure [subfig:imdb1689] is relatively more special. Its hidden relation consists of four terms rather than two. This means that it will show divergence if we cut three of the four propagators.

We have noticed that, while the preparation of this paper, a very interesting paper [78] was posted, studying the implications between spanning cuts inconsistency, magic relations, and the dimensions of critical varieties [83] from the Lee-Pomeransky polynomials. In this work, the authors include codes to find sectors where the critical varieties are with high dimensions, indicating that this implies the existence of magic relations, which usually breaks under the cuts. Although the algorithm in [78] and the Algorithm 2 in this paper are designed individually⁵ form different starting points, they appear to capture the same underlying mechanism. To specify this comment, we point out that in Algorithm 2, if a linear hidden relation is found, we know that the number of independent variables in the Lee-Pomenransky polynomials will be decreased. This leads to a higher-dimensioned critical variety. Thus, at least at linear level, this paper explains why the consistency problem is related to the dimension of critical variety. On the other hand, the studies in [78] gives us a very strong hint on how to find non-linear hidden relations.

5 Inconsistencies of external-massive diagrams↩︎

The examples in previous sections show that the diagrams with inconsistencies have a common feature. That is, they have the only one massless external line, or is factorized and has such diagram(s) as factor(s) (we call this feature 1EML for convenience in this paper). However, this does not mean that other diagrams are safe. In this section, we give an example with massive external lines, which still has inconsistency.

The diagram of the example is shown in Figure 4. The propagators of the example are \[P_1=l_1^2,\quad P_2=(l_1+p)^2-m^2,\quad P_3=l_2^2-s,\quad P_4=(l_1+l_2)^2-s,\quad P_5=(l_2+p)^2.\] \(P_5\) is an irreducible scalar product. The kinematic condition is \(p^2=s\), where \(s\neq m^2\).

For this example, we can write down an IBP relation as \[\label{eq:2L32IBP32with32h46t46} \begin{align} 0=&\int \frac{{\rm d}^{D}l_1}{i \pi^{\frac{D}{2}}} \frac{{\rm d}^{D}l_2}{i \pi^{\frac{D}{2}}}\frac{\partial}{\partial l_2^\mu}\frac{l_1^\mu}{D_1 D_2 D_3 D_4}\\ =&\int \frac{{\rm d}^{D}l_1}{i \pi^{\frac{D}{2}}} \frac{{\rm d}^{D}l_2}{i \pi^{\frac{D}{2}}}\frac{(D_3-D_4)(D_3+D_4-D_1)+i(\eta_3-\eta_4-\eta_1)D_3+i(\eta_3-\eta_4+\eta_1)D_4}{D_1 D_2 D_3^2 D_4^2}\\ =&({\mathcal{I}}_{1,1,0,2}-{\mathcal{I}}_{1,1,2,0}-{\mathcal{I}}_{0,1,1,2}+{\mathcal{I}}_{0,1,2,1}+i(\eta_3-\eta_4-\eta_1){\mathcal{I}}_{1,1,1,2}+i(\eta_3-\eta_4+\eta_1){\mathcal{I}}_{1,1,2,1}) \end{align}\tag{22}\] with hidden terms \[\label{eq:H34} {\mathcal{H}}_{34}=i(\eta_3-\eta_4)({\mathcal{I}}_{1,1,1,2}+{\mathcal{I}}_{1,1,2,1})-i\eta_1({\mathcal{I}}_{1,1,1,2}-{\mathcal{I}}_{1,1,2,1})\tag{23}\] If we apply \(\{1,3\}\text{-cut}\) on 22 and blindly erase the hidden terms proportional to \(\eta\)’s, we will get \[I_{1,1,2,0}|_{\{1,3\}\text{-cut}}=0\] However this is wrong. We can compute it as⁶ \[\label{eq:I1120321443-cut32direct32integration} \begin{align} I_{1,1,2,0}|_{\{1,3\}\text{-cut}}=&-\lim_{\eta_2\to{0^+}}(2\pi i)^2\int \frac{{\rm d}^D l_1{\rm d}^D l_2}{(i\pi^{\frac{D}{2}})^2}\frac{\delta(P_1)\delta^\prime(P_3)}{D_2}\\ =&4\pi e^{i \phi}\cot{(\frac{\pi D}{2})}\Gamma(3-D)\frac{ (s-m^2)^{D-3}}{s}\neq 0, \end{align}\tag{24}\] where \(\phi\) is a real parameter depending on the sign of \((s-m^2)\). This result shows that the inconsistency appears in this example.

Although we did not find the linear hidden relations, we still have an intuitive understanding for the cause of inconsistency of this example. Indeed, the diagram itself is not with feature 1EML. However, after cutting \(P_1\) (the massless propagator), making it on-shell, the sub-diagram (the massive bubble formed by \(P_3\) and \(P_4\)) is with the feature 1EML. We expect that under \(\{1\}-\)cut, \(P_3\) and \(P_4\) should be related by some relations, similar to the linear hidden relations defined in this paper.

There are still many works that can be done for this example. Firstly, one may want to reveal explicitly the relations between \(P_3\) and \(P_4\). Secondly, one may want to compute the hidden terms in this example to see whether it coincides 24 . We are leaving these works for future studies.

6 Summary↩︎

In this paper, we have identified one of the explanations of the spanning cuts consistency problem. We claim that, the conjecture IBP relations is unchanged under the cuts, is some times incorrect, if one blindly erases the hidden terms proportional to the “vanishing” Feynman prescription parameters.

We provided several examples to demonstrate this mechanism. In some of them, the inconsistencies can be explained by linear hidden relations (like those in Section 3 and Section 4), while the others cannot (like the one in Section 5). We have also observed that the inconsistencies usually appear together with the 1EML feature. This feature, in many cases, indicates the presence of hidden relations.

In Section 4, we provided an algorithm together with an implementation to find linear hidden relations. We have commented in Section 4 that, inspired by the very recent work [78] on this topic, the existence of hidden relations may be closely related to the dimensions of the critical varieties from the Lee-Pomeransky polynomial.

As a disclaimer, although this paper has demonstrated plenty of features that lead to inconsistency, we do not claim that these features exhaust all possible reasons of this problem. For example, we did not prove that diagrams without 1EML feature are safe. Nor have we proved that the families free of hidden relations are safe. Thus, there are still plenty of studies required on this topic. We leave these questions for future work.

Acknowledgement↩︎

We thank Bo Feng, Yuanche Liu, Yan-Qing Ma, Johann Usovitch, Ellis Ye Yuan and Yang Zhang for important discussions. Zihao Wu is supported by South China Normal University through grant No. 43102673. Yingxuan Xu is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under grant 396021762 - TRR 257.

7 Derivations↩︎

In this section, we collect the derivation details of some of the equations in this paper.

7.1 Equation 13↩︎

The derivation of equation 13 is to compute the integration of \({\mathcal{B}}_{12}\) defined in 11 . To make it convenient for the readers, we copy it here \[\label{eq:B1232appendix} {\mathcal{B}}_{12}= \frac{i \pi r_{12}}{2 r_1^2 r_2^2 (|{\vec{p}}\,|^2-r_{12}^2)} \Big(1+\frac{2 |{\vec{p}}\,|^2}{|{\vec{p}}\,|^2-r_{12}^2}-\frac{r_{12}^2}{r_1 r_2}\Big)\tag{25}\] The indefinite integration of \({\mathcal{B}}_{12}\) over \(l^1\) can be obtained, as⁷ \[\label{eq:IntB1232indefinite} \beta_{12}:={\int_{\text{id}}}{\mathcal{B}}_{12} {\rm d}l^1=-\frac{i \pi |{\vec{p}}\,|}{2 Q_{12}}\Big( \frac{3q_1+4(l^1)^2-F_{12}}{r_1 q_1}- \frac{3q_2+4(|{\vec{p}}\,|+l^1)^2-F_{12}}{r_2 q_2} \Big)\tag{26}\] where

\[\label{eq:rep32for32beta12} \begin{align} & q_1=|{\vec{z}}\,|^2+m^2+i \eta_1, \\& q_2=|{\vec{z}}\,|^2+m^2+i \eta_2, \\& Q_{12}=(\eta_1-\eta_2)^2+4 F_{12} |{\vec{p}}\,|^2,\\& F_{12}=q_1+i(\eta_1-\eta_2)\frac{ l^1}{|{\vec{p}}\,|}. \end{align}\tag{27}\]

In order to get the definite integration over \(l^1\), we need to analyze the poles and branch cuts of the integrand \({\mathcal{B}}_{12}\). In the denominator of the integrand, in apparent, \(|{\vec{p}}\,|^2-r_{12}^2\) contributes pole(s), and \(r_1\) and \(r_2\) contribute branch points and branch cuts.

7.1.0.1 Pole(s)

We analyze the pole(s). Let \[|{\vec{p}}\,|^2-r_{12}^2=0\] The L.H.S of the above equation contains a term with square roots \((-2 r_1 r_2)\). To remove it, consider \[\label{eq:A532Q12} (|{\vec{p}}\,|^2-r_{12}^2)( (|{\vec{p}}\,|^2-(r_1-r_2)^2))=-Q_{12}.\tag{28}\] Now, \(Q_{12}\) is linear in \(l^1\). For the equation \(Q_{12}=0\), if \(\eta_1=\eta_2\), there is no solution. Otherwise, there is a solution \[\label{eq:solution32Q12610} l^1=\frac{i}{4}\Big(\frac{\eta_1-\eta_2}{|{\vec{p}}\,|}+\frac{4|{\vec{p}}\,|(|{\vec{z}}\,|^2+m^2+ i \eta_1)}{\eta_1-\eta_2}\Big)\tag{29}\] This solution, of course, can make only one bracket vanish in the L.H.S. of 28 . To figure out which is the one, we substitute the solution 29 into \(r_1\) and \(r_2\), we have \[\label{eq:r132substituted32by32solution32of32Q12610} r_1=\frac{1}{4}\sqrt{-\frac{\Big((\eta_1-\eta_2)^2-4{|{\vec{p}}\,|}^2({|{\vec{z}}\,|}^2+m^2+i \eta_1)\Big)^2}{(\eta_1-\eta_2)^2{|{\vec{p}}\,|}^2}}\tag{30}\] and \[\label{eq:r232substituted32by32solution32of32Q12610} r_2=\frac{1}{4}\sqrt{-\frac{\Big((\eta_1-\eta_2)^2-4{|{\vec{p}}\,|}^2({|{\vec{z}}\,|}^2+m^2+i \eta_2)\Big)^2}{(\eta_1-\eta_2)^2{|{\vec{p}}\,|}^2}}.\tag{31}\] Since, by convention, the square root is with a branch cut along the negative real axis, \(r_1\) and \(r_2\) must have positive real parts. Thus, we have \[r_1={-\frac{(\eta_1-\eta_2)^2-4{|{\vec{p}}\,|}^2({|{\vec{z}}\,|}^2+m^2+i \eta_1)}{4i|\eta_1-\eta_2|{|{\vec{p}}\,|}}}\] and \[r_2={-\frac{(\eta_1-\eta_2)^2-4{|{\vec{p}}\,|}^2({|{\vec{z}}\,|}^2+m^2+i \eta_2)}{4i|\eta_1-\eta_2|{|{\vec{p}}\,|}}}.\] Obviously, \[r_1-r_2={|{\vec{p}}\,|}\text{Sign}(\eta_1-\eta_2)\] Thus, the solution 29 makes \[\label{eq:p942-40r1-r24194261032when32Q12610} {|{\vec{p}}\,|}^2-(r_1-r_2)^2=0.\tag{32}\] Since \(r_1r_2\neq 0\), this solution cannot make \({|{\vec{p}}\,|}^2-r_{12}^2=0\).

From the above, we know that the denominator \({|{\vec{p}}\,|}^2-r_{12}^2=0\) does not contribute poles for \({\mathcal{B}}_{12}\), both when \(\eta_1=\eta_2\) and \(\eta_1\neq \eta_2\).

7.1.0.2 Branch cuts

The branch cuts introduced by \(r_i\) can be described by \[r_i^2=-{\mathcal{R}}_i,\] where \(i=1,2\) and \({\mathcal{R}}_i\geq 0\) are real parameters describing the branch cuts and branch points (when \({\mathcal{R}}_i=0\)). Solving the above equations, we get the branch cuts from \(r_1\) are \[\label{eq:branch32cuts32r1} l^1=\pm\frac{1}{\sqrt{2}}\Big({\mathcal{K}}_1-\frac{i\eta_1}{{\mathcal{K}}_1}\Big),\tag{33}\] and the branch cuts from \(r_2\) are \[\label{eq:branch32cuts32r2} l^1=-{|{\vec{p}}\,|}\pm\frac{1}{\sqrt{2}}\Big({\mathcal{K}}_2-\frac{i\eta_2}{{\mathcal{K}}_2}\Big),\tag{34}\] where \[\label{eq:Ki32definition} {\mathcal{K}}_i=\sqrt{\sqrt{({|{\vec{z}}\,|}^2+m^2+{\mathcal{R}}_i)^2+\eta_i^2}-({|{\vec{z}}\,|}^2+m^2+{\mathcal{R}}_i)}\tag{35}\]

According to the above expressions, none of the branch cuts crosses the real axis. They are shown in Figure 5.

7.1.0.3 The integration

From the analysis of pole(s) and branch cuts above, we can now safely take the limit of \(\beta_{12}\) defined in 26 at \(l^1\to\pm\infty\), to get the definite integration of \({\mathcal{B}}_{12}\) along the real axis. At such limits, if \(\eta_1\neq\eta_2\), we have \[r_1\to |l^1|,\quad r_2\to |l^1|,\] and \[F_{12}\to i(\eta_1-\eta_2)\frac{l^1}{{|{\vec{p}}\,|}},\quad Q_{12}\to 4 i(\eta_1-\eta_2){|{\vec{p}}\,|}{l^1}.\] Thus, \[\label{eq:beta1232limit} \begin{align} \beta_{12}&\to-\frac{2 \pi i|{\vec{p}}\,|(l^1)^2}{ Q_{12}}\Big( \frac{1}{r_1 q_1}- \frac{1}{r_2 q_2} \Big)\\& \to -\frac{ \pi l^1}{2 (\eta_1-\eta_2)|l^1|}\Big( \frac{1}{q_1}- \frac{1}{ q_2} \Big) =\frac{i\pi}{2 q_1 q_2}{\text{Sign}}(l^1). \end{align}\tag{36}\] If \(\eta_1=\eta_2\), we have \[\label{eq:32noname32A4620} q_1=q_2=F_{12}, \quad Q_{12}=4 {|{\vec{p}}\,|}^2 q_1,\tag{37}\] We consider the factor in the bracket of \(\beta_{12}\) in 26 , as follows, \[\label{eq:beta1232factor32in32bracket32when32eta161eta2} \begin{align} &\frac{3q_1+4(l^1)^2-F_{12}}{r_1 q_1}- \frac{3q_2+4(|{\vec{p}}\,|+l^1)^2-F_{12}}{r_2 q_2}\\& =\Big(2+\frac{4(l^1)^2}{q_1}\Big)\Big(\frac{1}{r_1}-\frac{1}{r_2}\Big) - \frac{8{|{\vec{p}}\,|}l^1+4{|{\vec{p}}\,|}^2}{r_2 q_1}\\& =\Big(2+\frac{4(l^1)^2}{q_1}\Big)\frac{2{|{\vec{p}}\,|}l^1 +{|{\vec{p}}\,|}^2}{r_1 r_2r_{12}} - \frac{8{|{\vec{p}}\,|}l^1+4{|{\vec{p}}\,|}^2}{r_2 q_1}\\& =\frac{2{|{\vec{p}}\,|}l^1 +{|{\vec{p}}\,|}^2}{q_1r_1 r_2r_{12}}\Big(2q_1+4(l^1)^2-4r_1r_{12}\Big). \end{align}\tag{38}\] Since \[r_{12}:=r_1+r_2\to 2|l^1|,\] we have \[\frac{2{|{\vec{p}}\,|}l^1 +{|{\vec{p}}\,|}^2}{q_1r_1 r_2r_{12}}\Big(2q_1+4(l^1)^2-4r_1r_{12}\Big)\to-\frac{4(l^1)^3{|{\vec{p}}\,|}}{q_1|l^1|^3}=-\frac{4{|{\vec{p}}\,|}}{q_1}{\text{Sign}}(l^1).\] Using 26 and 37 , we have \[\label{eq:beta1232limit32when32eta161eta2} \beta_{12}\to\frac{i\pi }{2q_1^2}{\text{Sign}}(l^1).\tag{39}\] This agrees with 36 when taking \(\eta_1=\eta_2\). So, 36 applies also to the cases when \(\eta_1=\eta_2\). Using it, we can finally get \[\int {\mathcal{B}}_{12}{\rm d}l^l=\lim_{l^1\to \infty}\beta_{12}-\lim_{l^1\to -\infty}\beta_{12}=\frac{i\pi}{q_1q_2}.\] It is 13 .

7.2 Equation 16↩︎

We compute the integration of \({\mathcal{B}}_{{\underline{1}}2}\) defined in 14 . To make it convenient for the readers, we copy it here \[\label{eq:Bu1232appendix} {\mathcal{B}}_{{\underline{1}}2}= \frac{i \pi r_{{\underline{1}}2}}{2 r_{\underline{1}}^2 r_2^2 (|{\vec{p}}\,|^2-r_{{\underline{1}}2}^2)} \Big(1+\frac{2 |{\vec{p}}\,|^2}{|{\vec{p}}\,|^2-r_{{\underline{1}}2}^2}-\frac{r_{{\underline{1}}2}^2}{r_{\underline{1}}r_2}\Big)\tag{40}\] Its indefinite integration is

\[\label{eq:IntBu1232indefinite} \beta_{{\underline{1}}2}:={\int_{\text{id}}}{\mathcal{B}}_{{\underline{1}}2} {\rm d}l^1=-\frac{i \pi |{\vec{p}}\,|}{2 Q_{{\underline{1}}2}}\Big( \frac{3q_{\underline{1}}+4(l^1)^2-F_{{\underline{1}}2}}{r_{\underline{1}}q_{\underline{1}}}- \frac{3q_2+4(|{\vec{p}}\,|+l^1)^2-F_{{\underline{1}}2}}{r_2 q_2} \Big)\tag{41}\] where

\[\label{eq:rep32for32betau12} \begin{align} & q_{\underline{1}}=|{\vec{z}}\,|^2+m^2-i \eta_1, \\& q_2=|{\vec{z}}\,|^2+m^2+i \eta_2, \\& Q_{{\underline{1}}2}=(\eta_1+\eta_2)^2+4 F_{{\underline{1}}2} |{\vec{p}}\,|^2,\\& F_{{\underline{1}}2}=q_{\underline{1}}-i(\eta_1+\eta_2)\frac{ l^1}{|{\vec{p}}\,|}. \end{align}\tag{42}\]

Still, we analyze the pole(s) and branch cuts of \({\mathcal{B}}_{{\underline{1}}2}\).

7.2.0.1 Pole(s)

Similarly, we have \[\label{eq:A32Qu12} (|{\vec{p}}\,|^2-r_{{\underline{1}}2}^2)( (|{\vec{p}}\,|^2-(r_{\underline{1}}-r_2)^2))=-Q_{{\underline{1}}2},\tag{43}\] which is linear in \(l^1\). Since \(\eta_1+\eta_2>0\), the solution of \(Q_{{\underline{1}}2}=0\) is \[\label{eq:solution32Qu12610} l^1=-\frac{i}{4}\Big(\frac{\eta_1+\eta_2}{|{\vec{p}}\,|}+\frac{4|{\vec{p}}\,|(|{\vec{z}}\,|^2+m^2- i \eta_1)}{\eta_1+\eta_2}\Big)\tag{44}\] To figure out which bracket in the L.H.S. of 43 vanishes, we substitute 44 to \(r_{\underline{1}}\) and \(r_2\) to get \[\label{eq:ru132substituted32by32solution32of32Qu12610} r_{\underline{1}}=-\frac{1}{4}\sqrt{-\frac{\Big((\eta_1+\eta_2)^2-4{|{\vec{p}}\,|}^2({|{\vec{z}}\,|}^2+m^2-i \eta_1)\Big)^2}{(\eta_1+\eta_2)^2{|{\vec{p}}\,|}^2}}\tag{45}\] and \[\label{eq:r232substituted32by32solution32of32Qu12610} r_2=\frac{1}{4}\sqrt{-\frac{\Big((\eta_1+\eta_2)^2-4{|{\vec{p}}\,|}^2({|{\vec{z}}\,|}^2+m^2+i \eta_2)\Big)^2}{(\eta_1+\eta_2)^2{|{\vec{p}}\,|}^2}}.\tag{46}\] Also, considering the conventional branch cuts of the square root function, \(r_{\underline{1}}\) must have a negative real part, and \(r_2\) must have a positive one. Thus, we have \[r_1=-\frac{(\eta_1+\eta_2)^2-4{|{\vec{p}}\,|}^2({|{\vec{z}}\,|}^2+m^2-i \eta_1)}{4i(\eta_1+\eta_2){|{\vec{p}}\,|}}\] and \[r_2=-{\frac{(\eta_1+\eta_2)^2-4{|{\vec{p}}\,|}^2({|{\vec{z}}\,|}^2+m^2+i \eta_2)}{4i(\eta_1+\eta_2){|{\vec{p}}\,|}}}.\] Consequently, \[r_1-r_2=-{|{\vec{p}}\,|}\] Thus, the solution 44 makes \[\label{eq:p942-40r1-r24194261032when32Qu12610} {|{\vec{p}}\,|}^2-(r_{\underline{1}}-r_2)^2=0.\tag{47}\] Since \(r_{\underline{1}}r_2\neq 0\), this solution cannot make \({|{\vec{p}}\,|}^2-r_{{\underline{1}}2}^2=0\). i.e. the denominator \({|{\vec{p}}\,|}^2-r_{{\underline{1}}2}^2\) does not contribute any poles for \({\mathcal{B}}_{{\underline{1}}2}\)

7.2.0.2 Branch cuts

We describe the branch cuts introduced by \(r_i\) by \[r_i^2=-{\mathcal{R}}_i,\] where \(i={\underline{1}},2\) and \({\mathcal{R}}_i\geq 0\) are real parameters. The branch cuts from \(r_{\underline{1}}\) are \[\label{eq:branch32cuts32ru1} l^1=\pm\frac{1}{\sqrt{2}}\Big({\mathcal{K}}_{\underline{1}}+\frac{i\eta_1}{{\mathcal{K}}_{\underline{1}}}\Big),\tag{48}\] where \[\label{eq:Ku132definition} {\mathcal{K}}_{\underline{1}}=\sqrt{\sqrt{({|{\vec{z}}\,|}^2+m^2+{\mathcal{R}}_{\underline{1}})^2+\eta_{\underline{1}}^2}-({|{\vec{z}}\,|}^2+m^2+{\mathcal{R}}_{\underline{1}})}.\tag{49}\]

The branch cuts from \(r_2\) are already shown in 34 . In Figure 6 we draw the branch cuts of \({\mathcal{B}}_{{\underline{1}}2}\). None of them crosses the real axis. Thus, it is also safe to use \(\beta_{{\underline{1}}2}\) to compute the definite integration of \({\mathcal{B}}_{{\underline{1}}2}\).

7.2.0.3 The integration

Since \(\eta_1+\eta_2>0\), when \(l^1\to \infty\), we have \[r_{\underline{1}}\to -|l^1|, \quad r_2\to|l^1|,\] and \[F_{\underline{1}2}\to -i(\eta_1+\eta_2)\frac{l^1}{{|{\vec{p}}\,|}}, \quad Q_{\underline{1}2}\to -4i(\eta_1+\eta_2){l^1}{{|{\vec{p}}\,|}}\] Thus, \[\label{eq:betau1232limit} \begin{align} \beta_{{\underline{1}}2}&\to-\frac{2 \pi i|{\vec{p}}\,|(l^1)^2}{ Q_{{\underline{1}}2}}\Big( \frac{1}{r_{\underline{1}}q_{\underline{1}}}- \frac{1}{r_2 q_2} \Big)\\& \to\frac{2 \pi i|{\vec{p}}\,|(l^1)^2}{ Q_{{\underline{1}}2}|l^1|} \frac{q_{\underline{1}}+q_2}{ q_{\underline{1}}q_2}\\& \to-\frac{ \pi }{ 2(\eta_1+\eta_2)} \frac{q_{\underline{1}}+q_2}{ q_{\underline{1}}q_2}{\text{Sign}}(l^1). \end{align}\tag{50}\] Then, we have \[\int {\mathcal{B}}_{{\underline{1}}2}{\rm d}l^l=\lim_{l^1\to \infty}\beta_{{\underline{1}}2}-\lim_{l^1\to -\infty}\beta_{{\underline{1}}2}=-\frac{ \pi }{ \eta_1+\eta_2} \frac{q_{\underline{1}}+q_2}{ q_{\underline{1}}q_2}.\] It is 16 .

7.3 Equation 24↩︎

We compute \[\begin{align} {\mathcal{I}}_{1,1,2,0}|_{\{1,3\}\text{-cut}}=&-\int \frac{{\rm d}^D l_1{\rm d}^D l_2}{(i\pi^{\frac{D}{2}})^2}\frac{\delta(P_1)\delta^\prime(P_3)}{D_2} \end{align}\] Notice that \(P_1\) and \(D_2\) are free of \(l_2\), and \(P_3\) is free of \(l_1\), the expression is factorized as \[\begin{align} {\mathcal{I}}_{1,1,2,0}|_{\{1,3\}\text{-cut}}=\Big(2\pi i\int \frac{{\rm d}^D l_1}{i\pi^{\frac{D}{2}}}\frac{\delta(P_1)}{D_2}\Big)\Big(-2\pi i\int \frac{{\rm d}^D l_2}{i\pi^{\frac{D}{2}}}\delta^\prime(P_3)\Big) \end{align}\] For the integration over \(l_2\) in the second bracket, it is the same as what we computed in 6 (with \(m^2\) replaced by \(s\)). We can directly write it down as \[\begin{align} -2\pi i\int \frac{{\rm d}^D l_2}{i\pi^{\frac{D}{2}}}\delta^\prime(P_3)=-2s^{\frac{D-4}{2}}\Gamma(2-\frac{D}{2}) \end{align}\] We now compute the integrate over \(l_1\) in the first bracket. It is \[\label{eq:2L32int32over32l951940} \begin{align} &2\pi i\int \frac{{\rm d}^{D} l_1}{i\pi^{\frac{D}{2}}}\frac{\delta(P_1)}{D_2}\\ =&2\pi i\int \frac{{\rm d}^{D-1} {\vec{l}}_1{\rm d}l_1^0}{i\pi^{\frac{D}{2}}}\frac{\delta((l_1^0)^2-|{\vec{l}}_1|^2)}{(\sqrt{s}+l_1^0)^2 -|{\vec{l}}_1|^2-m^2-i \eta_2}\\ =&2\pi i\int \frac{{\rm d}^{D-1} {\vec{l}}_1}{i\pi^{\frac{D}{2}}}\frac{1}{2|{\vec{l}}_1|}\Big(\frac{1}{(\sqrt{s}+|{\vec{l}}_1|)^2 -|{\vec{l}}_1|^2-m^2-i \eta_2}+\frac{1}{(\sqrt{s}-|{\vec{l}}_1|)^2 -|{\vec{l}}_1|^2-m^2-i \eta_2}\Big)\\ =&2\pi i \int_0^{+\infty} \frac{{\rm d}|{\vec{l}}_1||{\vec{l}}_1|^{D-3}\Omega_{D-1}}{i\pi^{\frac{D}{2}}}\frac{s-m^2-i \eta_2}{(s-m^2-i \eta_2)^2-4 s |{\vec{l}}_1|^2}\\ =&\frac{2^{3-D}\pi^{\frac{3}{2}}s^{1-\frac{D}{2}}\csc{(\frac{\pi D}{2})} }{\kappa^3 \Gamma(\frac{D-1}{2})} \end{align}\tag{51}\] where \[\kappa:=s-m^2-i\eta_2.\] In the derivations in 51 , we have set \[p=(\sqrt{s},\vec{0}^{(D-1)}).\]

With the above, we have \[\label{eq:2L32direct32integration32before32eta2-620} {\mathcal{I}}_{1,1,2,0}|_{\{1,3\}\text{-cut}}=\frac{4\pi (i\kappa)^D\cot{(\frac{\pi D}{2})}\Gamma(3-D)}{s\kappa^3}.\tag{52}\] Taking \(\eta_2\to{0^+}\) we get 24 .

8 Checking the equations with Mathematica↩︎

The correctness of some equations (or statements) in this paper are checked using Mathematica. We provide the codes checking the equations in Derivations-1.wl in the GitHub repo.

Table 2 shows the correspondence of the sections in the code and the equations they check.

References↩︎

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1. See [29], [34], [75]–[77] for its applications.↩︎

2. For some ill cases, one should be careful. For example, in cases that the function multiplied with \(\delta\) function is singular at \(P_k\)=0, or it grows faster than \(|P_k|\) when \(|P_k|\to \infty\).↩︎

3. See Section 7.1 for its derivation.↩︎

4. See Section 7.2 for its derivation.↩︎

5. We hope this is convincing, because the submission dates of the two papers are very close.↩︎

6. See Appendix 7.3 for its derivation.↩︎

7. In this paper, we add a subscript “id” in the integration label, as \({\int_{\text{id}}}\), in order to emphasize that this is an indefinite integration.↩︎